Some theorems
@CihanPostsThms
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Posting some theorems, and occasionally other stuff. By @bahran_cihan
Joined July 2022
NON-THEOREM ANNOUNCEMENT: As someone struggling with depression/isolation, one reason I opened this account is to get some sense of belonging with the math community. I would appreciate if you can refrain from "that is obvious duh", "you miss the point" type of comments. Thanks.
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⌊e⌋ + ⌊φ⌋ = ⌊π⌋.
A simple and elegant formula relating e, π, i, and φ, the golden ratio.
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Given a finite group G, let Γ(G) be the graph whose vertices are conjugacy classes of non-central elements in G, where we connect distinct c,d if . gcd(|c|,|d|) ≠ 1 . [Bertram–Herzog–Mann 1990]. Given a finite group G, TFAE:.(1) Γ(G) has vertices, but no edges. (2) G ≅ S₃ .
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[Gruenhage–Schoenfeld 1975]. For an infinite compact metric space X, TFAE:. (1) The cardinality of nonempty open subsets of X up to homeomorphism is 2. (2) X is homeomorphic to the Cantor set.
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RT @CihanPostsThms: 𝐏𝐢𝐜𝐤'𝐬 𝐓𝐡𝐞𝐨𝐫𝐞𝐦: Let P be a polygon in ℝ² whose vertices lie in the lattice ℤ². Writing Int(P) for the interior of P and….
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[Wright 1954]. There are continuum many real numbers r such that the integer part of a (finite) power tower . 2^2^. ^2^r.is always prime.
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RT @jimfreitag: The cost of the military parade on June 14th is 1/5 of the 2024 NSF spending on mathematics research.
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It is conjectured that if m(G) = 3, then |G| ≤ |M|, where M is the Monster group.
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Given a finite group, let Irr(G) be the set of irreducible complex characters. For each d∈ℕ, write.m(G,d) := |{𝜒 ∈ Irr(G) : 𝜒(1) = d}|.and.m(G) := max{m(G,d) : d∈ℕ}. [Madrid 2025]. Let B be the Baby Monster group. If G is a finite group with m(G) = 2, then |G| ≤ |B|.
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[Gordon 2021]. There is no algorithm that decides whether a given PL 4-manifold is homeomorphic to the connected sum of 12 copies of S² x S². ----.A recent preprint of Tancer takes the 12 down to 9.
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The quotient ring. ℝ[x,y] / (x² + y² + 1) .is a PID which is not a Euclidean domain.
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In fact here S,g can be chosen such that. G = ⟨ S, gSg⁻¹ ⟩ .
[Aschbacher–Guralnick 1982]. For every finite group G, there exist. • a solvable subgroup S≤G,. • an element g∈G.such that G = ⟨S,g⟩.
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[Aschbacher–Guralnick 1982]. For every finite group G, there exist. • a solvable subgroup S≤G,. • an element g∈G.such that G = ⟨S,g⟩.
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[Suciu 1990]. Given integers r ≥ 1 and d ≥ 2r+3, there exist d-manifolds M and N which are homology d-spheres such that the following hold:. (1) The i-th homotopy groups of M and N are isomorphic for i≤r. (2) The (r+1)-st homotopy groups of M and N are not isomorphic.
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RT @CihanPostsThms: [Erdős 1964]: The following are equivalent:. (1) If there is a set Φ of entire functions such that {f(z) : f ∈ Φ} is co….
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Given a simplicial complex X with a specified edge e, write X/e for the space obtained by contracting e. [Carmesin–Lichev 2022]. There exists a 2-dimensional simply-connected simplicial complex X such that.• X embeds in ℝ³,.• X/e does not embed in ℝ³ for any edge e of X.
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